Spatiotemporal patterns in active four-state Potts models

Many types of spatiotemporal patterns have been observed under nonequilibrium conditions. Cycling through four or more states can provide specific dynamics, such as the spatial coexistence of multiple phases. However, transient dynamics have only been studied by previous theoretical models, since absorbing transition into a uniform phase covered by a single state occurs in the long-time limit. Here, we reported steady long-term dynamics using cyclic Potts models, wherein nucleation and growth play essential roles. Under the cyclic symmetry of the four states, the cyclic changes in the dominant phases and the spatial coexistence of the four phases are obtained at low and high flipping energies, respectively. Under asymmetric conditions, the spatial coexistence of two diagonal phases appears in addition to non-cyclic one-phase modes. The circular domains of the diagonal state are formed by the nucleation of other states, and they slowly shrink to reduce the domain boundary. When three-state cycling is added, competition between the two cycling modes changes the spatiotemporal patterns.